What is ln((cos^3(2x)sin^3(2x))/(e^{x^2+1)}) ?

The solution to ln((cos^3(2x)sin^3(2x))/(e^{x^2+1)}) is ln(cos^3(2x))+ln(sin^3(2x))-x^2-1

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simplify ln((cos^3(2x)sin^3(2x))/(e^{x^2+1)})

simplify

ln (\frac{cos ^{3} ( 2 x ) sin ^{3} ( 2 x )}{e ^{x^{2} + 1}})

ln (cos^{3} (2 x)) + ln (sin^{3} (2 x)) - x^{2} - 1

Solution steps

ln (\frac{cos ^{3} ( 2 x ) sin ^{3} ( 2 x )}{e ^{x^{2} + 1}})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{cos ^{3} ( 2 x ) sin ^{3} ( 2 x )}{e ^{x^{2} + 1}}) = ln (cos^{3} (2 x) sin^{3} (2 x)) - ln (e^{x^{2} + 1})

= ln (cos^{3} (2 x) sin^{3} (2 x)) - ln (e^{x^{2} + 1})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (e^{x^{2} + 1}) = (x^{2} + 1) ln (e)

= ln (cos^{3} (2 x) sin^{3} (2 x)) - (x^{2} + 1) ln (e)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (cos^{3} (2 x) sin^{3} (2 x)) = ln (cos^{3} (2 x)) + ln (sin^{3} (2 x))

= ln (cos^{3} (2 x)) + ln (sin^{3} (2 x)) - ln (e) (x^{2} + 1)

(x^{2} + 1) ln (e) = x^{2} + 1

= ln (cos^{3} (2 x)) + ln (sin^{3} (2 x)) - (x^{2} + 1)

- (x^{2} + 1) : - x^{2} - 1

= ln (cos^{3} (2 x)) + ln (sin^{3} (2 x)) - x^{2} - 1

4 lo g_{w} ((\frac{x}{1296})^{2}) - \frac{1}{3} lo g_{w} (2 x + 8)

\frac{1}{2} lo g_{b} (0.35) - \frac{1}{3} lo g_{b} (0.85)

\frac{9}{2} ln (60 - x) + ln (30 - x)

\frac{1}{1} - 5 ln (\frac{6}{5})

lo g_{5} (\frac{y ^{4}}{3 x})

What is ln((cos^3(2x)sin^3(2x))/(e^{x^2+1)}) ?
The solution to ln((cos^3(2x)sin^3(2x))/(e^{x^2+1)}) is ln(cos^3(2x))+ln(sin^3(2x))-x^2-1